Between reduced powers and ultrapowers, II.

Farah, Ilijas; Shelah, Saharon

doi:10.1090/tran/8777

Between reduced powers and ultrapowers, II.
HTML articles powered by AMS MathViewer

by Ilijas Farah and Saharon Shelah PDF

Trans. Amer. Math. Soc. 375 (2022), 9007-9034 Request permission

Abstract:

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.

References

James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, DOI 10.1016/0003-4843(79)90010-X
Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427. MR 2436146, DOI 10.1017/CBO9780511735219.011
Jörg Brendle and Tim LaBerge, Forcing tightness in products of fans, Fund. Math. 150 (1996), no. 3, 211–226. MR 1405043, DOI 10.4064/fm-150-3-211-226
J. Carrión, J. Gabe, C. Schafhauser, A. Tikuisis, and S. White, Classifying *-homomorphisms, Manuscript(s) in preparation.
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
H. G. Dales and W. H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series, vol. 115, Cambridge University Press, Cambridge, 1987. MR 942216, DOI 10.1017/CBO9780511662256
Alan Dow and Saharon Shelah, Asymmetric tie-points and almost clopen subsets of $\Bbb N^*$, Comment. Math. Univ. Carolin. 59 (2018), no. 4, 451–466. MR 3914712, DOI 10.14712/1213-7243.2015.268
R. Daniel Mauldin (ed.), The Scottish Book, 2nd ed., Birkhäuser/Springer, Cham, 2015. Mathematics from the Scottish Café with selected problems from the new Scottish Book; Including selected papers presented at the Scottish Book Conference held at North Texas University, Denton, TX, May 1979. MR 3242261, DOI 10.1007/978-3-319-22897-6
Ilijas Farah, Embedding partially ordered sets into $\omega _\omega$, Fund. Math. 151 (1996), no. 1, 53–95. MR 1405521
Ilijas Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702.
Ilijas Farah, How many Boolean algebras $\mathcal {P}(\mathbb {N})/\mathcal {I}$ are there?, Illinois J. Math. 46 (2003), 999–1033.
Ilijas Farah, Combinatorial set theory of C*-algebras, Springer Monographs in Mathematics, Springer, Cham, [2019] ©2019. MR 3971570, DOI 10.1007/978-3-030-27093-3
Ilijas Farah, Between reduced powers and ultrapowers, to appear in J. Eur. Math. Soc.
I. Farah, S. Ghasemi, A. Vaccaro, and A. Vignati, Corona rigidity, Preprint arXiv:2201.11618 (2022).
Ilijas Farah, Bradd Hart, Martino Lupini, Leonel Robert, Aaron Tikuisis, Alessandro Vignati, and Wilhelm Winter, Model theory of $\rm C^*$-algebras, Mem. Amer. Math. Soc. 271 (2021), no. 1324, viii+127. MR 4279915, DOI 10.1090/memo/1324
Ilijas Farah, Bradd Hart, and David Sherman, Model theory of operator algebras I: stability, Bull. Lond. Math. Soc. 45 (2013), no. 4, 825–838. MR 3081550, DOI 10.1112/blms/bdt014
Ilijas Farah, Bradd Hart, and David Sherman, Model theory of operator algebras II: model theory, Israel J. Math. 201 (2014), no. 1, 477–505. MR 3265292, DOI 10.1007/s11856-014-1046-7
Ilijas Farah and Paul McKenney, Homeomorphisms of Čech-Stone remainders: the zero-dimensional case, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2253–2262. MR 3767375, DOI 10.1090/proc/13736
Ilijas Farah and Saharon Shelah, A dichotomy for the number of ultrapowers, J. Math. Log. 10 (2010), no. 1-2, 45–81. MR 2802082, DOI 10.1142/S0219061310000936
Ilijas Farah and Saharon Shelah, Trivial automorphisms, Israel J. Math. 201 (2014), no. 2, 701–728. MR 3265300, DOI 10.1007/s11856-014-1048-5
Ilijas Farah and Saharon Shelah, Rigidity of continuous quotients, J. Inst. Math. Jussieu 15 (2016), no. 1, 1–28. MR 3427592, DOI 10.1017/S1474748014000218
S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103. MR 108455, DOI 10.4064/fm-47-1-57-103
Saeed Ghasemi, Reduced products of metric structures: a metric Feferman-Vaught theorem, J. Symb. Log. 81 (2016), no. 3, 856–875. MR 3569108, DOI 10.1017/jsl.2016.20
Kurt Gödel, What is Cantor’s continuum problem?, Amer. Math. Monthly 54 (1947), 515–525. MR 23780, DOI 10.2307/2304666
A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961/62), 123–128. MR 131986, DOI 10.4064/fm-50-2-123-128
Winfried Just and Adam Krawczyk, On certain Boolean algebras ${\scr P}(\omega )/I$, Trans. Amer. Math. Soc. 285 (1984), no. 1, 411–429. MR 748847, DOI 10.1090/S0002-9947-1984-0748847-1
Winfried Just and Žarko Mijajlović, Separation properties of ideals over $\omega$, Z. Math. Logik Grundlag. Math. 33 (1987), no. 3, 267–276. MR 894026, DOI 10.1002/malq.19870330311
Menachem Kojman and Saharon Shelah, Nonexistence of universal orders in many cardinals, J. Symbolic Logic 57 (1992), no. 3, 875–891. MR 1187454, DOI 10.2307/2275437
Sabine Koppelberg and Saharon Shelah, Subalgebras of Cohen algebras need not be Cohen, Logic: from foundations to applications (Staffordshire, 1993) Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 261–275. MR 1428008
Linus Kramer, Saharon Shelah, Katrin Tent, and Simon Thomas, Asymptotic cones of finitely presented groups, Adv. Math. 193 (2005), no. 1, 142–173. MR 2132762, DOI 10.1016/j.aim.2004.04.012
Kenneth Kunen, INACCESSIBILITY PROPERTIES OF CARDINALS, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (Ph.D.)–Stanford University. MR 2617841
Kenneth Kunen, Set theory, Studies in Logic (London), vol. 34, College Publications, London, 2011. MR 2905394
Georges Kurepa, L’hypothèse du continu et le problème de Souslin, Acad. Serbe Sci. Publ. Inst. Math. 2 (1948), 26–36 (French, with Serbian summary). MR 27818
Paul McKenney and Alessandro Vignati, Forcing axioms and coronas of $\mathrm {C}^*$-algebras, J. Math. Log. 21 (2021), no. 2, Paper No. 2150006, 73. MR 4290495, DOI 10.1142/S0219061321500069
L. Pacholski, On countably compact reduced products. III, Colloq. Math. 23 (1971), 5–15. MR 300888, DOI 10.4064/cm-23-1-5-15
L. Pacholski and C. Ryll-Nardzewski, On countably compact reduced products. I, Fund. Math. 67 (1970), 155–161. (errata insert). MR 265246, DOI 10.4064/fm-67-1-155-161
E. A. Paljutin, Categorical Horn classes. I, Algebra i Logika 19 (1980), no. 5, 582–614, 617 (Russian). MR 623786
M. Rørdam, Classification of nuclear, simple $C^*$-algebras, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR 1878882, DOI 10.1007/978-3-662-04825-2_{1}
Saharon Shelah, For what filters is every reduced product saturated?, Israel J. Math. 12 (1972), 23–31. MR 304157, DOI 10.1007/BF02764810
S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
Saharon Shelah, Vive la différence. I. Nonisomorphism of ultrapowers of countable models, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 357–405. MR 1233826, DOI 10.1007/978-1-4613-9754-0_{2}0
Saharon Shelah, Vive la différence. II. The Ax-Kochen isomorphism theorem, Israel J. Math. 85 (1994), no. 1-3, 351–390. MR 1264351, DOI 10.1007/BF02758648
Saharon Shelah, Toward classifying unstable theories, Ann. Pure Appl. Logic 80 (1996), no. 3, 229–255. MR 1402297, DOI 10.1016/0168-0072(95)00066-6
Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
Saharon Shelah, Vive la différence. III, Israel J. Math. 166 (2008), 61–96. MR 2430425, DOI 10.1007/s11856-008-1020-3
Saharon Shelah, No universal group in a cardinal, Forum Math. 28 (2016), no. 3, 573–585. MR 3510831, DOI 10.1515/forum-2014-0040
Saharon Shelah, Divide and conquer: dividing lines and universality, Theoria 87 (2021), no. 2, 259–348. MR 4329456, DOI 10.1111/theo.12289
Saharon Shelah and Juris Steprāns, Non-trivial automorphisms of $\scr {P}(\Bbb {N})/[\Bbb {N}]^{<\aleph _0}$ from variants of small dominating number, Eur. J. Math. 1 (2015), no. 3, 534–544. MR 3401904, DOI 10.1007/s40879-015-0058-0
Saharon Shelah and Alex Usvyatsov, Banach spaces and groups—order properties and universal models, Israel J. Math. 152 (2006), 245–270. MR 2214463, DOI 10.1007/BF02771986
Eric K. van Douwen, Kenneth Kunen, and Jan van Mill, There can be $C^*$-embedded dense proper subspaces in $\beta \omega -\omega$, Proc. Amer. Math. Soc. 105 (1989), no. 2, 462–470. MR 977925, DOI 10.1090/S0002-9939-1989-0977925-7
Eric K. van Douwen and Jan van Mill, Parovičenko’s characterization of $\beta \omega -\omega$ implies CH, Proc. Amer. Math. Soc. 72 (1978), no. 3, 539–541. MR 509251, DOI 10.1090/S0002-9939-1978-0509251-7
A. Vignati, Rigidity conjectures, Ann. Sci.Éc. Norm. Super. (to appear).
Wilhelm Winter, Structure of nuclear $\rm C^*$-algebras: from quasidiagonality to classification and back again, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1801–1823. MR 3966830
Jindřich Zapletal, Terminal notions, Bull. Symbolic Logic 5 (1999), no. 4, 470–478. MR 1792359, DOI 10.2307/421119